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# Chapter 9: Finance, growth and decay

## 9.1 Revision (EMBJD)

• Discuss terminology.
• Very important to emphasize not rounding off calculations until final answer.
• Learners should do calculation in one step using the memory function on their calculators.
• Draw timelines showing the different time periods, interest rates and any deposits/withdrawals.
• Discuss real-life financial issues; savings, budgets, tax, retirement etc.

Simple interest is the interest calculated only on the initial amount invested, the principal amount. Compound interest is the interest earned on the principal amount and on its accumulated interest. This means that interest is being earned on interest. The accumulated amount is the final amount; the sum of the principal amount and the amount of interest earned.

Formula for simple interest: $A = P(1 + in)$

Formula for compound interest: $A = P(1 + i)^n$ where \begin{align*} A &= \text{accumulated amount} \\ P &= \text{principal amount} \\ i &= \text{interest rate written as a decimal} \\ n &= \text{time period in years} \end{align*}

## Worked example 1: Simple and compound interest

Sam wants to invest $$\text{R}\,\text{3 450}$$ for $$\text{5}$$ years. Wise Bank offers a savings account which pays simple interest at a rate of $$\text{12,5}\%$$ per annum, and Grand Bank offers a savings account paying compound interest at a rate of $$\text{10,4}\%$$ per annum. Which bank account would give Sam the greatest accumulated balance at the end of the $$\text{5}$$ year period?

### Calculation using the simple interest formula

Write down the known variables and the simple interest formula

\begin{align*} P &= \text{3 450} \\ i &= \text{0,125} \\ n &= 5 \\ A &= P(1 + in) \end{align*}

Substitute the values to determine the accumulated amount for the Wise Bank savings account.

\begin{align*} A &= \text{3 450} (1 + \text{0,125} \times 5) \\ &= \text{R}\,\text{5 606,25} \end{align*}

### Calculation using the compound interest formula

Write down the known variables and the compound interest formula.

\begin{align*} P &= \text{3 450} \\ i &= \text{0,104} \\ n &= 5 \\ A &= P(1 + i)^n \end{align*}

Substitute the values to determine the accumulated amount for the Grand Bank savings account.

\begin{align*} A &= \text{3 450} (1 + \text{0,104})^5 \\ &= \text{R}\,\text{5 658,02} \end{align*}

The Grand Bank savings account would give Sam the highest accumulated balance at the end of the $$\text{5}$$ year period.

## Worked example 2: Finding $$i$$

Bongani decides to put $$\text{R}\,\text{30 000}$$ in an investment account. What compound interest rate must the investment account achieve for Bongani to double his money in $$\text{6}$$ years? Give your answer correct to one decimal place.

### Write down the known variables and the compound interest formula

\begin{align*} A &= \text{60 000} \\ P &= \text{30 000} \\ n &= 6 \\ A &= P(1 + i)^n \end{align*}

### Substitute the values and solve for $$i$$

\begin{align*} \text{60 000} &= \text{30 000} (1 + i)^6 \\ \frac{\text{60 000}}{\text{30 000}} &= (1 + i)^6 \\ 2 &= (1 + i)^6 \\ \sqrt{2} &= 1 + i \\ \sqrt{2} -1 &= i \\ \therefore i &= \text{0,122}\ldots \end{align*}

### Write the final answer and comment

We round up to a rate of $$\text{12,3}\%$$ p.a. to make sure that Bongani doubles his investment.

## Revision

Textbook Exercise 9.1

Determine the value of an investment of $$\text{R}\,\text{10 000}$$ at $$\text{12,1}\%$$ p.a. simple interest for $$\text{3}$$ years.

\begin{align*} A &= P(1 + in) \\ &= \text{10 000}(1 + \text{0,121} \times 3) \\ &= \text{R}\,\text{13 630} \end{align*}

Calculate the value of $$\text{R}\,\text{8 000}$$ invested at $$\text{8,6}\%$$ p.a. compound interest for $$\text{4}$$ years.

\begin{align*} A &= P(1 + i)^n \\ &= \text{8 000}(1 + \text{0,086})^3 \\ &= \text{R}\,\text{10 246,59} \end{align*}

Calculate how much interest John will earn if he invests $$\text{R}\,\text{2 000}$$ for $$\text{4}$$ years at:

$$\text{6,7}\%$$ p.a. simple interest

\begin{align*} A &= P(1 + in) \\ &= \text{2 000}(1 + \text{0,067} \times 4) \\ &= \text{R}\,\text{2 536} \end{align*}

$$\text{5,4}\%$$ p.a. compound interest

\begin{align*} A &= P(1 + i)^n \\ &= \text{2 000}(1 + \text{0,054})^4 \\ &= \text{R}\,\text{2 468,27} \end{align*}

The value of an investment grows from $$\text{R}\,\text{2 200}$$ to $$\text{R}\,\text{3 850}$$ in $$\text{8}$$ years. Determine the simple interest rate at which it was invested.

\begin{align*} A &= P(1 + in) \\ \text{3 850} &= \text{2 200}(1 + 8i) \\ \frac{\text{3 850}}{\text{2 200}} - 1 &= 8i \\ \therefore \frac{1}{8} \left( \frac{\text{3 850}}{\text{2 200}} - 1 \right) &= i \\ \therefore i &= \text{9,38}\% \end{align*}

James had $$\text{R}\,\text{12 000}$$ and invested it for $$\text{5}$$ years. If the value of his investment is $$\text{R}\,\text{15 600}$$, what compound interest rate did it earn?

\begin{align*} A &= P(1 + i)^n \\ \text{15 000}&= \text{12 000}(1 + i)^5 \\ \frac{\text{15 000}}{\text{12 000}} &= (1 + i)^5 \\ \sqrt{\frac{\text{15 000}}{\text{12 000}}} - 1 &= i \\ \therefore i &= \text{4,56}\% \end{align*}